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Numerische Mathematik

, Volume 70, Issue 3, pp 303–310 | Cite as

Continuous bordering of matrices and continuous matrix decompositions

  • Willy Govaerts
  • Bodo Werner

Summary.

Let \({\cal M}_n\) be the set of all real \(n\times n\)-matrices of rank \(\ge n-1\). We prove that for \(n\ge 2\) there are no continuous vector fields \(l,r:{\cal M}_n\rightarrow {\Bbb R}^n\) such that the bordered matrix \(\) is regular for all \(A\in{\cal M}_n\). This result has some relevance for the numerical analysis of steady state bifurcation. As a by-product we show that there is no nonvanishing continuous vector field \(u:{\cal M}_n^{(n-1)}\rightarrow {\Bbb R}^n\) with \(Au(A)=0\) for all \(A\in{\cal M}_n^{(n-1)}\), where \({\cal M}_n^{(n-1)}\subset {\cal M}_n\) is the set of all matrices of rank deficiency one. This implies that there is no singular value decomposition of \(A\) depending continuously on \(A\) in any matrix set which contains \({\cal M}_n^{(n-1)}\). As another application we prove that in general there is no global analytic singular value decomposition for analytic matrix valued functions of more than one real variable.

Mathematics Subject Classification (1991):15A18 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Willy Govaerts
    • 1
  • Bodo Werner
    • 2
  1. 1.Belgian National Fund of Scientific Research N.F.W.O., Department of Applied Mathematics and Computer Science, Krijgslaan 281, B-9000 Gent, Belgium BE
  2. 2.Institut für Angewandte Mathematik der Universität Hamburg, Bundesstrasse 55, D-20146 Hamburg, Germany DE

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